One of the difficulties with the Mapper is however the choice of parameters such as deciding on the overlap between the bins and design of the filtering function \(f\) . These choices determine the construction of the resulting graph, and which filtering function that best captures the topology is not always entirely clear. While the choice of function allows for certain freedom, it might also pose as a barrier of entry for the regular practitioner and a difficulty in generating useful representations when using the Mapper algorithm. The optimal is of course to reduce the dimensionality in a way such that the local and the global topology of the data is preserved, while the summarization of the data lose the minimum amount of information. However, as [1]} points out, different filtering functions capture different things. [2]} mention that the two most common choices are the eccentricity, i.e., the maximum distance to any other point in the point cloud, and the eigenfunctions of the covariance matrix as used in PCA. Other methods such as SVD ([3]}) as well as the kernel density estimation ([4]}) have been extensively used. These do however all come with their drawbacks. SVD and PCA do not capture non-linear relations very well and might miss important non-linear properties of the data, although they preserve distance. Non-linear techniques such as t-SNE ([5]}) and IsoMAP ([6]}) are better at preserving local distances but lose the global perspective as pointed out by [1]}. The kernel density estimator relies on a bandwidth \(\epsilon \) , set by the user, introducing another parameter to tune. The choice of filtering function is therefore a choice for the user where trade-offs have to be made. Finding a filtering function without the aforementioned drawbacks and which could simultaneously capture both the global and local topologies would therefore be highly beneficial, since it would produce Mapper graphs representing the shape of the data more accurately.
